Solving Quadratic Equations — Challenge
Four methods, one goal. Factoring is fastest when it works; the quadratic formula always works.
What these problems cover
These questions use 10 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Area — Space inside, measured in square units.
- Perfect square — A number that is some whole number squared: 1, 4, 9, 16, 25, …
- Square root — For k ≥ 0, √k is the nonnegative number whose square is k.
- Factor — A number that divides evenly into another.
- Parabola — The U-shaped graph of a quadratic function.
- Roots — The x values where y = 0.
- Zero product property — If AB = 0 then A = 0 or B = 0.
- Quadratic formula — x = -b ± √b2 - 4ac2a.
- Discriminant — b2 - 4ac. Its sign counts the real solutions.
- Completing the square — Rewriting x2 + bx as a perfect square minus a constant.
Problem 1
What is the positive solution of x2=49?
- 14
- 7
- 24.5
- 49
- 2401
Worked answer
7
Taking square roots of both sides gives x = 7 or x = -7.
The positive one is 7.
Halving 49 would give 24.5, which confuses squaring with doubling.
Problem 2
What is the larger solution of x2-5x+6=0?
- 1
- 2
- 3
- 5
- 6
Worked answer
3
Factorise: two numbers multiplying to 6 and adding to -5 are -2 and -3, so (x-2)(x-3) = 0.
A product is zero only when a factor is zero, so x = 2 or x = 3.
The larger solution is 3.
Problem 3
What is the positive solution of x2+2x-8=0?
- -4
- 1
- 2
- 4
- 8
Worked answer
2
Two numbers multiplying to -8 and adding to 2 are 4 and -2, so (x+4)(x-2) = 0.
The solutions are x = -4 and x = 2.
The positive one is 2.
Problem 4
What is the positive solution of x2-9=0?
- 4.5
- 3
- 9
- 81
- -3
Worked answer
3
Rearranging gives x2 = 9.
So x = 3 or x = -3, and the positive solution is 3.
Equivalently, factorising as (x-3)(x+3) = 0 gives the same two roots.
Problem 5
What is the positive solution of 2x2=18?
- 6
- 9
- 3
- 36
- 4.5
Worked answer
3
Divide both sides by 2 first: x2 = 9.
Then x = 3 or x = -3, so the positive solution is 3.
Taking the square root of 18 before dividing is the common error.
Problem 6
What is the solution of x2+6x+9=0?
- 3
- -3
- -9
- 9
- No real solution
Worked answer
-3
The left side is a perfect square: x2+6x+9 = (x+3)2.
So (x+3)2 = 0, which gives x+3 = 0 and x = -3.
There is only one solution here, because the two roots coincide.
Problem 7
What is the sum of the solutions of x2-4x+3=0?
- 2
- 3
- 4
- -4
- 7
Worked answer
4
Factorising gives (x-1)(x-3) = 0, so the solutions are 1 and 3.
Their sum is 1+3 = 4.
This matches the general rule that the roots of x2+bx+c sum to -b, here -(-4) = 4.
Problem 8
What is the product of the solutions of x2+x-12=0?
- 12
- -12
- -1
- 1
- 7
Worked answer
-12
Two numbers multiplying to -12 and adding to 1 are 4 and -3, so (x+4)(x-3) = 0.
The solutions are -4 and 3, and their product is -12.
This matches the rule that the roots of x2+bx+c multiply to c.